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20 December 2017

New gcse: ratio.

ratio problem solving bbc bitesize

  • Mel from JustMaths collated ratio Higher GCSE questions from sample and specimen papers here , and has written up her solutions here . 
  • If you subscribe to MathsPad then you'll be pleased to hear that they have lovely resources for ratio including a set of questions for Higher GCSE  with loads of examples like the problems I've featured in this post.  
  • Don Steward has plenty of ratio tasks  including his set of ' Harder Ratio Questions ' and a really helpful collection of GCSE ratio and proportion questions .
  • On MathsBot you can generate ratio questions, revision grids and practice papers. Select 'ratio, proportion and rates of change' at the top.
  • There are exam style questions in this collection from Lucy Kilgariff on TES.
  • OCR has a 'Calculations with Ratio' Topic Check In  and AQA has a Ratio and Proportion Topic Test .
  • David Morse of Maths4Everyone has shared a set of revision exercises and ratio exam style questions .

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20 comments:

ratio problem solving bbc bitesize

My ratio pages don't get much attention - not sure why since I think they're instructive and easy to use. They don't support the particular type of harder questions described in the post (but I'll look to add something along those lines), but they do help understanding the concept of a ratio and it's utility. Manipulation of ratio quantities: http://thewessens.net/ClassroomApps/Main/ratios.html?topic=number&path=Main&id=7 Introduction to the ratio concept: http://www.thewessens.net/blog/2015/03/19/ratios-the-super-fractions/ Bar model visualisation of ratios: http://thewessens.net/ClassroomApps/Models/BarModels/visualfractionratio.html?topic=models&path=Models&id=17 Multiplicative word problems: http://thewessens.net/ClassroomApps/Models/BarModels/multiplicationwordproblems.html?topic=models&path=Models&id=8 Ken

ratio problem solving bbc bitesize

Fantastic! Thanks Ken.

ratio problem solving bbc bitesize

Thanks so much for your blog on ratio question types. Although I've been a maths teacher/tutor for over 30 year, ratio has always been a bug bear for me. I could wing it with old style gcse because I learnt the types of solutions required, however I have been stressed on the new types. This blog has made me think through ratios and I am certainly a lot happier. Bryan

Excellent, I'm so pleased it helps.

On your fractions approach, a quick trick is to realise that a/c = a/b x b/c. Makes it quite quick to work out (That is, if the students are good with cancelling down when multiplying). However, what I find confuses students about writing ratios as fractions is that it confuses the part:part idea of a ratio with the part:whole idea of a fraction. Perhaps that's why it's somewhat counter-intuitive. Also, final point is that ratios (fractions) and vectors is another application of harder ratio questions that often pops up on the new GCSE. Thanks for the post, Farah

Thanks for the comment!

This is a fabulous resource on work that is missing from the new GCSE texts that I have seen. Lovely challeging questions to make students think.

Thanks! Glad it's helpful.

I've been using equivalent ratios for these type of questions. Find what doesn't change - the total number of sweets. Write ratios as equivalent ratios where the parts that doesn't change are the same. 3:7 has 10 parts, 3:5 has 8 parts LCM of 8 and 10 is 40 Ratios are 12:28 and 15:25 Number of sweets given is 3. Also works for following question Ratio of blue to red counters in a bag is 1:2, I add 12 blue counters and the ratio is 5:7. How many red counters are in the bag? What doesn't change? Red counters LCM of 2 and 7 is 14 Ratios are 7:14 and 10:14 3 parts are 12 counters, 1 part is 4 counters and 14 parts are 56 red counters. Also Jill is 4 times older than Jack. In 14 years time the ratio of Jack's age to Jill is 5:7. How old is Jill now? Ratios are 1:4 and 5:7. What doesn't change? The difference between their ages Find two equivalent ratios where difference between them is the same. 4 - 1 = 3 and 7 - 5 = 2 LCM of 2 and 3 is 6 Equivalent ratios are 2:8 and 15:21 13 parts = 26 years, 1 part is 2 years, Jill is 16.

This is the approach I use. I think it's logical.

Thank you! Yes, this is logical. Same approach as bar modelling (but without the visual).

Oops, my mistake, third example should be .....in 26 years time the ratio of their ages is 5:7 ..... I did try to represent these using bar modelling at first but struggled to find a model that was intuitive and actually helped with the question. I would be grateful if anyone has ideas on this.

Although some bar modelling experts would disagree, I don't think bar modelling is intuitive/helpful for harder ratio questions. Bar modelling is fantastic for easier ratio questions, but when the questions get more complicated it's often really hard to figure out how to draw the scenario - definitely not as easy as some people make out!

Thank you for the post. Brilliant as usual. I actually did the sweets question in my class once. I simply said that Alice fraction of sweets changed from 7/10 to 5/8 when she gave the 3 sweets away. If we just subtract those fractions, the fraction remaining, 7/10 - 5/8 = 3/40. This means that Alice originally had 40 sweets.

Hadnt considered tis method but I love it

Thanks Stephen. I guess it makes sense, as the fraction lost is equivalent to the 3 sweets divided by the total.

Love this! Thanks for sharing.

Hi Jo, thanks for the post which I came across via a tweet you put out a couple of days ago - which also tied in with a question and the same method I saw in my step-daughters book the very next day - freaky! It is a more compact method than I would normally use in my teaching and will be switching to it. I think the only tweak I might make is to write the algebra ratio above the numeric one so the starting fractions are (7x-3)/5 : (3x+3)/3 The reason being that some students might get a little scared seeing algebra as part of the denominator but less so when faced with a number.

Good idea - thank you!

Hi Jo, One method I use when teaching questions like the first one above (Alice gives 3 sweets to Olivia) is the following. To begin with Alice has 7/10 of the sweets and then after giving three to Olivia, her share has reduced to 5/8 of the sweets. So Alice's share has reduced by (7/10 - 5/8=) 3/40 which is equivalent to 3 sweets, therefore there must be 40 sweets in total. Students can then proceed in answering the relevant question. I must admit I only use this method with the top sets.

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Ratio and Proportion KS2

This collection is one of our Primary Curriculum collections - tasks that are grouped by topic.

ratio problem solving bbc bitesize

Pumpkin Pie Problem

Peter wanted to make two pies for a party. His mother had a recipe for him to use. However, she always made 80 pies at a time. Did Peter have enough ingredients to make two pumpkin pies?

Rectangle Tangle

The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the shapes?

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In the Money

One quarter of these coins are heads but when I turn over two coins, one third are heads. How many coins are there?

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Orange Drink

A 750 ml bottle of concentrated orange squash is enough to make fifteen 250 ml glasses of diluted orange drink. How much water is needed to make 10 litres of this drink?

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Fraction Fascination

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

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After training hard, these two children have improved their results. Can you work out the length or height of their first jumps?

ratio problem solving bbc bitesize

ratio problem solving bbc bitesize

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Solving Ratio Problems

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  • We add the parts of the ratio to find the total number of parts.
  • There are 2 + 3 = 5 parts in the ratio in total.
  • To find the value of one part we divide the total amount by the total number of parts.
  • 50 ÷ 5 = 10.
  • We multiply the ratio by the value of each part.
  • 2:3 multiplied by 10 gives us 20:30.
  • The 50 counters are shared into 20 counters to 30 counters.

videolesson.JPG

  • 2 + 3 = 5 and so there are 5 parts in the ratio in total.
  • We divide by this total number of parts to find the value of each part.
  • We multiply the original ratio by the value of each part.
  • We have 20:30.

videolesson.JPG

  • Sharing in a Ratio: Part 1

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Ratio Problems: Worksheets and Answers

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How to Solve Ratio Problems

Share £50 in the ratio 2:3

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Key Stage 3 Maths - Lesson Objectives, Keywords and Resources - Year 8 - Number

  • Cut-the-knot
  • Curriculum Online
  • Starter of the Day

Lesson Objectives

To be able to:

  • Reduce a ratio to its simplest form, including a ratio expressed in different units, recognising links with fraction notation;
  • Divide a quantity into two or more parts in a given ratio
  • Use the unitary method to solve simple word problems involving ratio and direct proportion.
  • Compare two ratios.

ratio, unitary method, percentage, fraction, conversion

Home | Year 7 | Year 8 | Year 9 | Starter of the Day | Links

They may then be asked to estimate the percentage of the mixture that is sultanas, for instance.

Solve problems involving shapes and a scale factor

Your child will solve problems involving ‘similar shapes’ . These are shapes with the same proportions that are not necessarily in the same orientation or the same size. For example:

ratio problem solving bbc bitesize

In this instance, your child might only be given the value of one of the sides of the larger shape. They may then be asked to figure out the value of the other two sides.

Your child will be expected to solve problems where the scale factor  between two shapes is known or can be found. Scale factor means the amount by which you increase the size of the shape. For example, in the picture above there is a scale factor of 2, which you can work out by comparing the lengths of the sides.

Solve ratio problems involving unequal sharing and grouping

Your child will use their knowledge of fractions , division , and multiplication to solve unequal sharing or grouping problems. They may be asked to solve problems such as:

Lisa and Stephanie share £360. Lisa gets £40 more than Stephanie. How much does Stephanie get?   One way to solve this is to halve £360 to get £180, halve £40 to get £20, and then add them together to get Lisa’s cut of the money, £200. This leaves Stephanie with £160 .

frac{2}{5}

Find possibile combinations of two variables in an equation

Your child will be able to find numerical solutions to equations where two of the values are unknown.

For example, your child should be able to find the values of a and b that make this equation correct: 2 a + b = 28. Children will use trial and error to explore different possibilities :

In the equation 2 a  +  b  = 28, you may take the value of  a  to be 4. If  a  = 4, then 2 a  = 8. So, 8 +  b  = 28. Therefore,  b  = 28 – 8. So,  b  = 20.
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How to Calculate Ratios

Last Updated: January 29, 2024 References

This article was co-authored by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 3,082,260 times.

Ratios are mathematical expressions that compare two or more numbers. They can compare absolute quantities and amounts or can be used to compare portions of a larger whole. Ratios can be calculated and written in several different ways, but the principles guiding the use of ratios are universal to all.

Practice Problems

ratio problem solving bbc bitesize

Understanding Ratios

Step 1 Be aware of how ratios are used.

  • Ratios can be used to show the relation between any quantities, even if one is not directly tied to the other (as they would be in a recipe). For example, if there are five girls and ten boys in a class, the ratio of girls to boys is 5 to 10. Neither quantity is dependent on or tied to the other, and would change if anyone left or new students came in. The ratio merely compares the quantities.

Step 3 Notice the different ways in which ratios are expressed.

  • You will commonly see ratios represented using words (as above). Because they are used so commonly and in such a variety of ways, if you find yourself working outside of mathematic or scientific fields, this may the most common form of ratio you will see.
  • Ratios are frequently expressed using a colon. When comparing two numbers in a ratio, you'll use one colon (as in 7 : 13). When you're comparing more than two numbers, you'll put a colon between each set of numbers in succession (as in 10 : 2 : 23). In our classroom example, we might compare the number of boys to the number of girls with the ratio 5 girls : 10 boys. We can simply express the ratio as 5 : 10.
  • Ratios are also sometimes expressed using fractional notation. In the case of the classroom, the 5 girls and 10 boys would be shown simply as 5/10. That said, it shouldn't be read out loud the same as a fraction, and you need to keep in mind that the numbers do not represent a portion of a whole.

Using Ratios

Step 1 Reduce a ratio...

  • In the classroom example above, 5 girls to 10 boys (5 : 10), both sides of the ratio have a factor of 5. Divide both sides by 5 (the greatest common factor) to get 1 girl to 2 boys (or 1 : 2). However, we should keep the original quantities in mind, even when using this reduced ratio. There are not 3 total students in the class, but 15. The reduced ratio just compares the relationship between the number of boys and girls. There are 2 boys for every girl, not exactly 2 boys and 1 girl.
  • Some ratios cannot be reduced. For example, 3 : 56 cannot be reduced because the two numbers share no common factors - 3 is a prime number, and 56 is not divisible by 3.

Step 2 Use multiplication or...

  • For example, a baker needs to triple the size of a cake recipe. If the normal ratio of flour to sugar is 2 to 1 (2 : 1), then both numbers must be increased by a factor of three. The appropriate quantities for the recipe are now 6 cups of flour to 3 cups of sugar (6 : 3).
  • The same process can be reversed. If the baker needed only one-half of the normal recipe, both quantities could be multiplied by 1/2 (or divided by two). The result would be 1 cup of flour to 1/2 (0.5) cup of sugar.

Step 3 Find unknown variables when given two equivalent ratios.

  • For example, let's say we have a small group of students containing 2 boys and 5 girls. If we were to maintain this proportion of boys to girls, how many boys would be in a class that contained 20 girls? To solve, first, let's make two ratios, one with our unknown variables: 2 boys : 5 girls = x boys : 20 girls. If we convert these ratios to their fraction forms, we get 2/5 and x/20. If you cross multiply, you are left with 5x=40, and you can solve by dividing both figures by 5. The final solution is x=8.

Grace Imson, MA

Grace Imson, MA

Look at the order of terms to figure out the numerator and denominator in a word problem. The first term is usually the numerator, and the second is usually the denominator. For example, if a problem asks for the ratio of the length of an item to its width, the length will be the numerator, and width will be the denominator.

Catching Mistakes

Step 1 Avoid addition or subtraction in ratio word problems.

  • Wrong method: "8 - 4 = 4, so I added 4 potatoes to the recipe. That means I should take the 5 carrots and add 4 to that too... wait! That's not how ratios work. I'll try again."
  • Right method: "8 ÷ 4 = 2, so I multiplied the number of potatoes by 2. That means I should multiply the 5 carrots by 2 as well. 5 x 2 = 10, so I want 10 carrots total in the new recipe."

Step 2 Convert...

  • A dragon has 500 grams of gold and 10 kilograms of silver. What is the ratio of gold to silver in the dragon's hoard?

{\frac  {1,000grams}{1kilogram}}

  • The dragon has 500 grams of gold and 10,000 grams of silver.

{\frac  {500gramsGold}{10,000gramsSilver}}={\frac  {5}{100}}={\frac  {1}{20}}

  • Example problem: If you have six boxes, and in every three boxes there are nine marbles, how many marbles do you have?

6boxes*{\frac  {3boxes}{9marbles}}=...

One common problem is knowing which number to use as a numerator. In a word problem, the first term stated is usually the numerator and the second term stated is usually the denominator. If you want the ratio of the length of an item to the width, length becomes your numerator and width becomes your denominator.

Community Q&A

Community Answer

You Might Also Like

Calculate Compression Ratio

  • ↑ http://www.virtualnerd.com/common-core/grade-6/6_RP-ratios-proportional-relationships/A
  • ↑ http://www.purplemath.com/modules/ratio.htm
  • ↑ http://www.helpwithfractions.com/math-homework-helper/least-common-denominator/
  • ↑ http://www.mathplanet.com/education/algebra-1/how-to-solve-linear-equations/ratios-and-proportions-and-how-to-solve-them
  • ↑ http://www.math.com/school/subject1/lessons/S1U2L2DP.html

About This Article

Grace Imson, MA

To calculate a ratio, start by determining which 2 quantities are being compared to each other. For example, if you wanted to know the ratio of girls to boys in a class where there are 5 girls and 10 boys, 5 and 10 would be the quantities you're comparing. Then, put a colon or the word "to" between the numbers to express them as a ratio. In this example, you'd write "5 to 10" or "5:10." Finally, simplify the ratio if possible by dividing both numbers by the greatest common factor. To learn how to solve equations and word problems with ratios, scroll down! Did this summary help you? Yes No

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Ratio and proportion

Unit 9 – 2 weeks

The PowerPoint file contains slides you can use in the classroom to support each of the learning outcomes for this unit, listed below. The slides are comprehensively linked to associated pedagogical guidance in the  NCETM Primary Mastery Professional Development materials . There are also links to the ready-to-progress criteria detailed in the  DfE Primary Mathematics Guidance 2020 .

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Unitary Method of Solution (F)

Unitary Method of Solution (F)

The Unitary Method sounds like it might be complicated but it’s not! It’s a very useful way to solve problems involving ratio and proportion, so once you’ve mastered the technique in this GCSE Maths quiz have a go at the Ratio and Proportion quiz too!

The method involves scaling down one of the variables to a single unit, i.e. 1. Once we know the value of 1 unit, the value of multiple units can be found by multiplying. This will make more sense when it is explained using an example. If 12 tins of paint weigh 30kg, how much will 5 tins weigh? The first step in solving this is to find what ONE tin weighs. This will be 30/12 so 2.5kg. Scaling this back up for 5 tins gives 5 * 2.5 = 12.5kg.

The unitary method is also useful for working out which deal will give better value for money.

A 300ml can of juice costs 85p, while a 500ml bottle of the same juice costs £1.20. Which is better value?

As you are going to be in a shop when faced with this problem, you probably won’t have your calculator on you to divide by 300 and 500! Rather than working out what 1ml will cost, it is just as valid to work out the cost of 100ml for each option. For the can, 100ml will cost (85/3) 28.33 pence, whilst for the bottle 100ml will cost (120/5) 24 pence. You should be able to perform these calculations in your head: with practise, you will improve! We can see in this comparison that the bottle is better value.

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In this introduction to ratio, children will learn what a ratio is and how to simplify ratios. They can then work through some ratio word problems, with step-by-step workings and answers given on every slide.

  • Key Stage: Key Stage 2
  • Subject: Maths
  • Topic: Ratio and Proportion
  • Topic Group: Ratio and Proportion
  • Year(s): Year 6
  • Media Type: PowerPoint
  • Resource Type: Front-of-Class Teaching
  • Last Updated: 05/10/2022
  • Resource Code: M2PAT112
  • Curriculum Point(s): Solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts.

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Golden Ratio

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618

It appears many times in geometry, art, architecture and other areas.

The Idea Behind It

Have a try yourself (use the slider):

This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?

Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

Do you think it is the "most pleasing rectangle"?

Maybe you do or don't, that is up to you!

parthenon golden ratio

Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way.

The Actual Value

The Golden Ratio is equal to:

1.61803398874989484820... (etc.)

The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number , and I will tell you more about it later.

We saw above that the Golden Ratio has this property:

a b = a + b a

We can split the right-hand fraction then do substitutions like this:

a b = a a + b a ↓      ↓      ↓ φ =  1 + 1 φ

So the Golden Ratio can be defined in terms of itself!

Let us test it using just a few digits of accuracy:

With more digits we would be more accurate.

Powers (Exponents)

Let's try multiplying by φ :

φ = 1 + 1 φ ↓     ↓     ↓ φ 2 = φ + 1

That ended up nice and simple. Let's multiply again!

φ 2 = φ + 1 ↓     ↓     ↓ φ 3 = φ 2 + φ

The pattern continues! Here is a short list:

Note how each power is the two powers before it added together! The same idea behind the Fibonacci Sequence (see below).

Calculating It

You can use that formula to try and calculate φ yourself.

First guess its value, then do this calculation again and again:

  • A) divide 1 by your value (=1/value)
  • C) now use that value and start again at A

With a calculator, just keep pressing "1/x", "+", "1", "=", around and around.

I started with 2 and got this:

It gets closer and closer to φ the more we go.

But there are better ways to calculate it to thousands of decimal places quite quickly.

Here is one way to draw a rectangle with the Golden Ratio:

  • Draw a square of size "1"
  • Place a dot half way along one side
  • Draw a line from that point to an opposite corner
  • Now turn that line so that it runs along the square's side
  • Then you can extend the square to be a rectangle with the Golden Ratio!

(Where did √5 2 come from? See footnote*)

A Quick Way to Calculate

That rectangle above shows us a simple formula for the Golden Ratio.

When the short side is 1 , the long side is 1 2 + √5 2 , so:

φ = 1 2 + √5 2

The square root of 5 is approximately 2.236068, so the Golden Ratio is approximately 0.5 + 2.236068/2 = 1.618034. This is an easy way to calculate it when you need it.

Interesting fact : the Golden Ratio is also equal to 2 × sin(54°) , get your calculator and check!

Fibonacci Sequence

There is a special relationship between the Golden Ratio and the Fibonacci Sequence :

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

(The next number is found by adding up the two numbers before it.)

And here is a surprise: when we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio .

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation.

Let us try a few:

We don't have to start with 2 and 3 , here I randomly chose 192 and 16 (and got the sequence 192, 16,208,224,432,656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ... ):

The Most Irrational

I believe the Golden Ratio is the most irrational number . Here is why ...

So, it neatly slips in between simple fractions.

Note: many other irrational numbers are close to rational numbers, such as Pi = 3.14159265... is pretty close to 22/7 = 3.1428571...)

No, not witchcraft! The pentagram is more famous as a magical or holy symbol. And it has the Golden Ratio in it:

  • a/b = 1.618...
  • b/c = 1.618...
  • c/d = 1.618...

Read more at Pentagram .

Other Names

The Golden Ratio is also sometimes called the golden section , golden mean , golden number , divine proportion , divine section and golden proportion .

Footnotes for the Keen

* where did √5/2 come from.

With the help of Pythagoras :

c 2 = a 2 + b 2

c 2 = ( 1 2 ) 2 + 1 2

c 2 = 1 4 + 1

c = √( 5 4 )

Solving using the Quadratic Formula

We can find the value of φ this way:

Which is a Quadratic Equation and we can use the Quadratic Formula:

φ = −b ± √(b 2 − 4ac) 2a

Using a=1 , b=−1 and c=−1 we get:

φ = 1 ± √(1+ 4) 2

And the positive solution simplifies to:

Kepler Triangle

That inspired a man called Johannes Kepler to create this triangle:

It is really cool because:

  • it has Pythagoras and φ together
  • the ratio of the sides is 1 : √φ : φ , making a Geometric Sequence .

bias research results

bias research results

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    That rectangle above shows us a simple formula for the Golden Ratio. When the short side is 1, the long side is 1 2+√5 2, so: φ = 1 2 + √5 2. The square root of 5 is approximately 2.236068, so the Golden Ratio is approximately 0.5 + 2.236068/2 = 1.618034. This is an easy way to calculate it when you need it.

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